Question

In a binomial distribution, the mean is 4 and the variance is 3. Then the number of trials $n$ is:

• A. 8
• B. 12
• C. 16
• D. 20

Hint:

In a binomial distribution, the variance is always less than the mean. The ratio $\text{Variance}/\text{Mean}$ gives $q$ directly, so $p = 1 - q$, and then $n = \text{Mean}/p$.

Answer 1

The Correct Option is C

Step 1: Concept
For a binomial distribution with $n$ trials and probability of success $p$, the Mean is given by $np$ and the Variance is given by $npq$, where $q = 1 - p$ is the probability of failure.

Step 2: Meaning
We are given the Mean ($np = 4$) and Variance ($npq = 3$). We can divide the variance by the mean to find $q$, then find $p$, and finally calculate $n$.

Step 3: Analysis
Given: \[ np = 4 \] \[ npq = 3 \] Dividing Variance by Mean: \[ \frac{npq}{np} = \frac{3}{4} \implies q = \frac{3}{4} \] Since $p + q = 1$: \[ p = 1 - q = 1 - \frac{3}{4} = \frac{1}{4} \] Substitute $p$ into the Mean equation: \[ n \cdot \left(\frac{1}{4}\right) = 4 \implies n = 16 \]

Step 4: Conclusion
The number of trials $n$ is 16. Final Answer: (C)